What is (\partial)/(\partial y)(xe^{xy}+y^2sin(xy)) ?

The answer to (\partial)/(\partial y)(xe^{xy}+y^2sin(xy)) is x^2e^{xy}+2ysin(xy)+cos(xy)xy^2

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(\partial)/(\partial y)(xe^{xy}+y^2sin(xy))

\frac{\partial}{\partial y} (x e^{x y} + y^{2} sin (x y))

x^{2} e^{x y} + 2 y sin (x y) + cos (x y) x y^{2}

Solution steps

\frac{\partial}{\partial y} (x e^{x y} + y^{2} sin (x y))

Treat

x

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial y} (x e^{x y}) + \frac{\partial}{\partial y} (y^{2} sin (x y))

\frac{\partial}{\partial y} (x e^{x y}) = x^{2} e^{x y}

\frac{\partial}{\partial y} (y^{2} sin (x y)) = 2 y sin (x y) + cos (x y) x y^{2}

= x^{2} e^{x y} + 2 y sin (x y) + cos (x y) x y^{2}

What is (\partial)/(\partial y)(xe^{xy}+y^2sin(xy)) ?
The answer to (\partial)/(\partial y)(xe^{xy}+y^2sin(xy)) is x^2e^{xy}+2ysin(xy)+cos(xy)xy^2